Time Step Rescaling Resolves the Lagging Effect in Stabilization Single-Step Schemes for a Class of L² Gradient Flows

Stabilization techniques offer significant advantages in developing highly stable algorithms for gradient flows; however, the considerable lagging effect arising from different integrations of stabilization terms continues to pose a major challenge. For a class of L² gradient flows with derivative-independent nonlinearity, we propose a unified framework to analyze the rescaled time step and the associated enlargement of the time-step constraint in a class of stabilized single-step schemes. As examples, we consider the implicit-explicit Runge–Kutta schemes, and exponential-time-differencing Runge–Kutta schemes. A unified matrix-vector framework is developed to establish their energy stability. We first demonstrate that the unstabilized scheme can maintain the energy dissipation under a specific time-step constraint. In contrast, the stabilized scheme exhibits energy dissipation for any time step, provided that the stabilization parameter is sufficiently large. By reformulating these stabilized single-step schemes into a class of explicit Runge–Kutta integrators, we characterize the time delay introduced by stabilization and eliminate the lagging phenomenon through a relaxation technique. The maximum rescaled time step shows a considerable enlargement of the time-step constraint compared to the unstabilized scheme. Numerical experiments confirm the delay-free behavior, efficiency, and energy dissipation property of proposed schemes.